During a lecture at UCLA, Serge Lang asked what is the most important function in mathematics. This question is quite personal and every person certainly has his own opinion. In fact, a professor spoke out loud that the constant function is the most important function.

Serge Lang suggested . The function is **self-dual** (i.e. its Fourier transform is itself, see below) and the property is essential to the **functional equation of the Riemann zeta function**.

For , the **Fourier transform**of is a function on defined by

Since , the above integral is absolutely convergent.

Theorem 1The function is self-dual, i.e., .

*Proof 1.*We will use Cauchy’s theorem from the theory of complex analysis. For another proof without the theory of complex analysis, see proof 2 below.

We will show that the integral on the right hand side is and completes the proof. Let be the contour of the rectangle with vertices , , , . By Cauchy’s theorem

Let , the second term and the fourth term . Thus

The last equality is a standard application of double integration in polar coordinates.

Before we proceed to the second proof, we need the following lemma.

Lemma 2Suppose .

(i)If has a continuous derivative and , then

(ii) Let . Suppose is integrable, then

*Proof*. Suppose

In the last step we use the fact that and hence . This proves (i). Next Suppose is integrable. Then is also integrable. Hence

Thus (ii) follows.

*Proof 2*. Let . Simple calculation shows that

Applying the Fourier transform to the above equation, by the above lemme we obtain

Solving the ordinary differential equation, we obtain

where is a constant. To find , we set